Method and apparatus for void content measurement and method and apparatus for particle content measurement

ABSTRACT

A void or particle content is determined using the X-ray small angle scattering measurement for a sample made of a thin film having voids or particles disorderly dispersed in the matrix, the diffraction peaks being not available for such a sample. The invention includes three aspects. The first aspect is that an equipment constant is determined and an unknown void or particle content is calculated based on the equipment constant. The second aspect is that a plurality of samples having unknown matrix densities are prepared, the matrix densities are determined so that differences in the matrix densities among the samples become a minimum, and a void or particle content is calculated based on the matrix density and the scale factor of the X-ray small angle scattering. The third aspect is for a plurality of samples having unknown particle densities, and executes procedures similar to those of the second aspect.

CROSS-REFERENCE TO RELATED APPLICATION

The present application is a Divisional Application of U.S. applicationSer. No. 11/072,924 filed Mar. 4, 2005, now U.S. Pat. No. 7,272,206which is incorporated herein by reference.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a method and an apparatus for themeasurement of a void content and a particle content with the use of theX-ray small angle scattering method.

2. Description of the Related Art

The present invention relates to the measurement of the void content orthe particle content of a sample made of a thin film which has a matrixand voids or particles dispersed in the matrix. The matrix representsthe base material in which voids or particles exist. FIGS. 1A and 1B areexemplary sectional views of the sample to which the present inventionis applied. The sample shown in Fig. IA is a thin film 10 on a substrate18, the thin film 10 having a matrix 12 and voids 14 dispersed in thematrix 12. Thinking about the specific region of the thin film, a ratioof the total volume of the voids 14 existing in the region to the volumeof the thin film in the region can be defined as a void content in theregion. The sample shown in FIG. 1B is a thin film 10 having a matrix 12and particles 16, whose material is different from the matrix, dispersedin the matrix 12. Similarly to the case having voids, thinking about thespecific region of the thin film, a ratio of the total volume of theparticles 16 existing in the region to the volume of the thin film inthe region can be defined as a particle content in the region.

A technique of forming particles or voids with a nanometer size in athin film gets a lot of attention in the recent development of thenanotechnology. The particles with a nanometer size are in the spotlightmainly in view of the improvement and variation of the properties causedby the quantum size effect. The voids with a nanometer size are expectedto realize the porous interlayer insulation material in connection withthe fine structure wiring of the semiconductor device. The particles orvoids with a nanometer size can not be observed by the ordinary X-raydiffraction method because of the small periodicity. Therefore, theX-ray small angle scattering method and the EXAFS method would beimportant for observing the electron density fluctuation with ananometer order. Especially, the X-ray small angle scattering method hasbeen used from old times as the technique for evaluating the electrondensity fluctuation in a material with several nanometers to severalhundred nanometers, for example, it has been used for the sizeevaluation of the particles or the voids and the evaluation of thelong-period structure.

It is noted that the present invention relates to the measurement of thevoid content or the particle content of the sample with the use of theX-ray small angle scattering method, such a measurement is disclosed inJapanese patent publication No. 2001-349849 A, which will be referred toas the first publication. The first publication discloses the analysisof the thin film having voids or particles dispersed therein with theuse of the X-ray small angle scattering method and the parameter fittingoperation between the measured profile of the scattered intensity andthe theoretical profile of the scattered intensity to determine theoptimum values of the parameters. The embodiment of the firstpublication uses the scattering function for the theoretical profile ofthe scattered intensity, the function being a model in which the voidsor the particles are assumed to be spherical or cylindrical and the sizeand its variance (which indicates the distribution of the size) are usedas parameters to determine the size and the variance of the void or theparticle. The first publication also discloses that a scatteringfunction is produced using a void or particle content and itscorrelation distance as parameters, and a parameter fitting operation iscarried out to determine the void or particle content and itscorrelation distance.

Formulae for determining the scattered X-ray intensity in connectionwith a thin film having voids or particles are disclosed in Japanesepatent publication No. 2003-202305 A, which will be referred to as thesecond publication.

It is noted that the embodiment of the present invention carries out themeasurement of the average density and the film thickness of the thinfilm using the X-ray reflectance method as the preliminary step beforeproducing the theoretical profile of the X-ray small angle scattering,the measurement of the average density and the film thickness of thethin film using the X-ray reflectance method is known and disclosed in,for example, Japanese patent publication No. 10-38821 A (1998), whichwill referred to as the third publication.

Although the prior art disclosed in the first publication can measurethe void or particle content using the X-ray small angle scatteringmethod, the prior art has the problems described below. The methoddisclosed in the first publication uses a model in which the analysis iseffective in the case that the proximal distance of the voids orparticles is held at a certain distance, i.e., the state of theshort-range-order. When the proximal distance of the voids or particlesis held at a certain distance, a diffraction peak corresponding to thedistance can be observed on the X-ray small angle scattering pattern.The proximal distance of the voids or particles and the void or particlecontent can be evaluated based on the appearance angle of thediffraction peak, which corresponds to the proximal distance, and thespread of the diffraction peak, which is evaluated with thefull-width-at-half-maximum and corresponds to the void or particlecontent. This method is effective only in the case that the proximaldistance is held at a certain distance and thus is not applicable to asystem in which voids or particles are randomly dispersed.

SUMMARY OF THE INVENTION

It is an object of the present invention to provide a method and anapparatus for measurement which can determine a void content or aparticle content with the use of the X-ray small angle scattering methodeven when the diffraction peak can not be observed.

The present invention discloses three aspects. The first aspect is asfollows. An equipment constant of X-ray small angle scattering equipmentis determined for a sample having a known void or particle content.Unknown void or particle content of another sample is calculated usingthe equipment constant.

The second aspect is as follows. There are prepared a plurality ofsamples which have unknown matrix densities expected to be identicalwith each other and void or particle contents expected to be differentfrom each other. The scale factor of the X-ray small angle scattering isdetermined for each of the samples. The matrix density of each of thesamples is determined using the scale factors under the provision that“the difference in matrix density among the samples becomes a minimum”.The void or particle content is calculated based on the determinedmatrix densities and the scale factors. In measuring the void contentaccording to the second aspect, the particle density should be known.

The third aspect is as follows. There are prepared a plurality ofsamples made of thin films which have known matrix densities, unknownparticle densities expected to be identical with each other, andparticle contents expected to be different from each other. The scalefactor of the X-ray small angle scattering is determined for each of thesamples. The particle density of each of the samples is determined usingthe scale factors under the provision that “the difference in particledensity among the samples becomes a minimum”. The particle content iscalculated based on the determined particle densities and the scalefactors.

Any aspect of the invention utilizes the scale factor, which expressesthe absolute value of the X-ray small angle scattering, as an importantelement and the void or particle content is calculated based on thescale factor. The three aspects of the invention are common to eachother in the viewpoint of the use of the scale factor.

The present invention has the advantages described below. The void orparticle content can be determined with the use of the X-ray small anglescattering measurement even when the voids or particles are randomlydispersed and thus the diffraction peaks can not be observed. With theuse of the first aspect, the equipment constant can be calculated usingsamples having known void or particle contents, and thereafter unknownvoid or particle contents can be determined using the equipmentconstant. With the use of the second and the third aspects, theequipment constant can be determined using samples having different voidor particle contents even without a sample having a known void orparticle content, and thereafter unknown void or particle contents canbe determined using the equipment constant.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1A and 1B are sectional views of samples;

FIG. 2 shows formulae (1) to (7) in connection with calculation of avoid content or a particle content;

FIG. 3 shows formulae (8) to (14) in connection with calculation of avoid content or a particle content;

FIG. 4 shows formulae (15) to (17) in connection with calculation of avoid content or a particle content;

FIG. 5 shows formulae (18) to (20) in connection with calculation of avoid content or a particle content;

FIG. 6 shows formulae (21) to (24) in connection with calculation of avoid content or a particle content;

FIGS. 7A to 7C are explanatory views of X-ray reflectance measurementand X-ray small angle scattering measurement;

FIG. 8 is a graph showing results of the X-ray reflectance measurement;

FIG. 9 shows a table indicating parameters determined in the X-rayreflectance method;

FIG. 10 is a graph showing an incident X-ray intensity;

FIG. 11 is a graph showing results of the offset scanning in the X-raysmall angle scattering measurement;

FIG. 12 is a graph showing profile fitting for the offset scanning inthe X-ray small angle scattering measurement;

FIG. 13 is a graph showing profile fitting for the rocking scanning inthe X-ray small angle scattering measurement;

FIG. 14 shows a table indicating parameters and scale factors determinedin the X-ray small angle scattering method;

FIG. 15 is a graph showing relative matrix densities among the samples;

FIG. 16 is a graph showing relative void contents among the samples;

FIG. 17 shows a table indicating values used for calculating voidcontents and calculated void contents;

FIG. 18 shows formulae (25) and (26) expressing scattering functions;

FIG. 19 shows a formula (27) expressing atomic scattering factor and atable indicating an example of calculation for the average atomicscattering factor and the average atomic mass;

FIG. 20 shows formulae (28) to (34) in connection with calculation of avoid content or a particle content;

FIG. 21 shows formulae (35) and (36) in connection with calculation of avoid content or a particle content;

FIG. 22 shows formulae (37) to (41) in connection with calculation of avoid content or a particle content; and

FIG. 23 shows formulae (42) to (46) in connection with calculation of avoid content or a particle content.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Embodiments of the present invention will now be described below withreference to the drawings. Before entering the explanation of theembodiments of the present invention, a method for calculating a voidcontent will be described in the case of a thin film having a knownmatrix density. When a thin film having no voids and another thin filmhaving voids can be prepared and the matrices of these thin films arethe same material, i.e., the same matrix density, the void content ofthe thin film having voids can be calculated as described below. First,the density of the thin film having no voids is measured by the X-rayreflectance method, the result being a matrix density ρ_(M). Next, thedensity of the thin film having voids is measured by the X-rayreflectance method, the result being an average density ρ_(F) of thethin film having voids. Expressing a void density as ρ_(pore) and a voidcontent as p, formula (1) in FIG. 2 is effected. Since the void densityρ_(pore) is zero, the void content p becomes formula (2) in FIG. 2.Then, if the matrix density ρ_(M) is determined by the X-ray reflectancemethod, the void content can be calculated with a high accuracy.

When particles are dispersed in the matrix in place or the voids it isneeded only to exchange the void density ρ_(pore) to a particle densityρ_(par) in formula (1), the result being shown as formula (34) in FIG.20. In this case, if the particle density ρ_(par) is known, the particlecontent p can be calculated with the use of the average density ρ_(F) ofthe thin film having particles, which can be measured by the X-rayreflectance method, and the density ρ_(M) of the thin film having noparticles, which can be also measured by the X-ray reflectance method ifsuch a thin film is prepared.

Although the above-described method for calculating the void content orthe particle content when the matrix density is known is not included inthe present invention, such an explanation would become the basicknowledge for understanding the present invention.

The first aspect of the present invention will now be described. Thepoint of the first aspect is that when a thin film having a known voidcontent or particle content can be prepared, an equipment constant of anX-ray small angle scattering measurement equipment can be determinedusing such a thin film and then a void content or particle content of athin film having an unknown void content or particle content can bedetermined based on the equipment constant. Thinking about a thin filmhaving voids, there occurs a scattered X-ray caused by the difference indensity of electrons between the matrix and the void, the scatteredX-ray intensity I being expressed by formula (3) in FIG. 2, where k₀ isthe equipment constant, p is the void content, r_(e) is the radius of aclassical electron orbit, f_(F) is an average atomic scattering factorof the thin film, M_(f) is an average atomic mass of the thin film,ρ_(F) is an average density of the thin film, and F is a scatteringfunction. The scattering function F is a function of a scattering vectorq and a parameter “para”. The scattering vector q depends on an X-rayincident angle against the thin film, an X-ray outgoing angle from thethin film and an X-ray wavelength used. The parameter “para” depends ona scattering function model. The form of the scattering function f willbe described later. The equipment constant k₀ is a coefficient dependingon the brightness of the incident X-ray, a geometric arrangement of theequipment, a slit size and so on, it being a specific value inherent inthe equipment.

In formula (3) in FIG. 2, the scattered X-ray intensity I on the leftside is a value actually detected with an X-ray detector, whereas a partbetween two absolute-value symbols on the right side can be calculatedusing a theoretical formula. Stating in detail, the average atomicscattering factor f_(F), the average atomic mass M_(f) and the averagedensity ρ_(F) may be numerical values. The scattering function F may becalculated with variable parameters. The average density ρ_(F) may be avalue measured by the X-ray reflectance method, or may be apredetermined value if it is known by any other method. Accordingly, anunknown term in formula (3) is k₀p/(1−p) only, which is that determinedas a result of the profile fitting operation as described below, itbeing expressed by S and being called as a scale factor. Then, formula(3) in FIG. 2 becomes formula (4), and the scale factor S in formula (4)is expressed by formula (5). Formula (5) is the basic formula indicatinga relationship among the scale factor S, the equipment constant k₀ andthe void content p. Transforming formula (5) so that the equipmentconstant k₀ moves on the left side, formula (6) is effected. When thescale factor S is determined by the X-ray small angle scattering methodfor a thin film having a known void content p, the right side of formula(6) can be calculated and the equipment constant k₀ can be determined.When the equipment constant k₀ is determined, the void content p can becalculated, as seen from formula (7), using the scale factor S and theequipment constant k₀ after the scale factor S was determined by theX-ray small angle scattering method for a thin film having an unknownvoid content p. The description above is the principle of calculation ofthe void content according to the first aspect of the present invention.

Incidentally, formula (4) mentioned above should be, strictly speaking,expressed by formulae (37) to (41) in FIG. 22. Namely, the scatteredintensity I is a sum of four scattered intensities I_(a) to I_(d). Theintensity I_(a) is one caused by a phenomenon in which an incident X-rayis scattered by the voids or particles in the thin film. The intensityI_(b) is one caused by a phenomenon in which the incident X-ray whichhas been scattered by the voids or particles is further reflected at theboundary. The intensity I_(c) is one caused by a phenomenon in which theincident X-ray which has been reflected at the boundary is furtherreflected at the voids or particles. The intensity I_(d) is one causedby a phenomenon in which the incident X-ray which has been reflected atthe boundary and further reflected at the voids or particles is furtherreflected at the boundary, i.e., multiple reflections. The totalintensity I becomes a scattered intensity I in the thin film havingvoids or particles. Explaining symbols appearing in formulae (38) to(41), q_(L) ⁺ is a scattering vector and is expressed by formula (42) inFIG. 23. The symbol q_(L) ⁻ is also a scattering vector and is expressedby formula (43). The symbol v₀ is a wavenumber vector and is expressedby formula (44). The symbol α_(L) is a refraction angle at the incidenceand is expressed by formula (45). The symbol ζ_(L) is a refraction angleat the outgoing and is expressed by formula (46). The symbol Imrepresents the imaginary part of a complex number. The symbol d_(L) isthe thickness of a thin film, the film thickness value determined by theX-ray reflectance method being substituted for d_(L). The suffix Lrepresents the Lth layer. Stating in detail, when the thin film is madeof a multilayer, the scattered intensity is determined for each layer,L=1, 2, 3, . . . , and the L intensities are summed up. It is noted thatalthough the scattering function F is a function of the parameter“para”, it is omitted in expression in formulae (38) to (41).

Referring to formulae (42) and (43) in FIG. 23, the symbol Re representsreal part of a complex number. The symbol λ in formula (44) is thewavelength of the incident X-ray. The symbol θ₀ in formula (45) is anincident angle of an X-ray which is incident on the thin film. Thesymbol n_(L) is a refraction index of the thin film, which can be easilycalculated from the average density of the thin film. The averagedensity may be an average density ρ_(F) determined by the X-rayreflectance measurement. The symbol φ₀ in formula (46) is an outgoingangle of an X-ray which outgoes from the thin film.

The determination of the scattered intensity for the thin film havingvoids or particles as shown in formulae (37) to (46) is known anddisclosed in the second publication mentioned above, for example.

A method for determining a particle content according to the firstaspect will now be described. Thinking about a thin film havingparticles, there occurs a scattered X-ray caused by the difference indensity of electrons between the matrix and the particle, the scatteredX-ray intensity I being expressed by formula (8) in FIG. 3, wheref_(par) is an average atomic scattering factor of the particle, M_(par)is an average atomic mass of the particle, ρ_(par) is an average densityof the particle, and other symbols are the same as in formula (3)Formula (8) can be transformed to formula (9) using the scale factor S.If the particle density ρ_(par) is known, the scale factor S can bedetermined by the profile fitting operation in the X-ray small anglescattering method similarly to the case having the voids. The scalefactor S has the same form as formula (5) mentioned above. Accordingly,when the scale factor S is determined by the X-ray small anglescattering method for the thin film having a known particle content p,the equipment constant k₀ can be determined. When the equipment constantk₀ is determined, the particle content p can be calculated using thescale factor S and the equipment constant k₀ after the scale factor Swas determined by the X-ray small angle scattering method for a thinfilm having an unknown particle content p. The description above is theprinciple of calculation of the particle content according to the firstaspect of the present invention.

It is noted that formula (8) is based on that the particle densityρ_(par) is known, formula (8) is not applicable to the case that theparticle density is unknown. However, if the matrix density ρ_(M) isknown in such a case, a scattered X-ray intensity can be determinedusing formula (28) in FIG. 20. Formula (28) can be transformed toformula (29) using the scale factor S. Thus, if the matrix density ρ_(M)is known, the scale factor S can be determined by the profile fittingoperation in the X-ray small angle scattering method similarly to thecase having voids, the scale factor S having the form of formula (30).Accordingly, when the scale factor S is determined by the X-ray smallangle scattering method for the thin film having a known particlecontent p, the equipment constant k₀ can be determined. When theequipment constant k₀ is determined, the particle content p can becalculated, as seen from formula (32), using the scale factor S and theequipment constant k₀ after the scale factor S was determined by theX-ray small angle scattering method for a thin film having an unknownparticle content p. The description above is another method ofcalculation of the particle content according to the first aspect of thepresent invention.

The second aspect of the present invention will now be described. Thesecond aspect is used in the case that a thin film having no voids isnot available. In this case, here are prepared a plurality of sampleswhich have matrix densities expected to be identical with each otherwhereas void contents expected to be different from each other.Generally speaking, a void content p_(i) of the ith sample is expressedby formula (10) in FIG. 3, ρ_(Fi) being an average density of the ithsample and ρ_(Mi) being a matrix density of the ith sample. Thescattered X-ray intensity I(i) caused by the difference in density ofelectrons between the matrix and the void can be expressed by formula(11) in FIG. 3, this formula having the same form as formula (3) exceptthe difference of having the symbol indicating the ith number. Formula(11) can be transformed to formula (12) using the scale factor S_(i)which is be expressed by formula (13).

Since the equipment constant k₀ does not depend on the sampler arelational expression shown in (14) is effected based on formula (13).Namely, there is effected a relational expression between the scalefactor S and the void content p among plural samples. Since the voidcontent p can be expressed using the average density ρ_(F) and thematrix density ρ_(M) of the sample as shown in formula (2) in FIG. 2,formula (2) is applied to formula (14) to effect formula (15). Further,formula (16) is effected based on formula (14), and formula (17) iseffected based on formula (15).

Referring to formula (16), the void content p₁ of a sample 1 isexpressed by the scale factor S₁ of the sample 1, the scale factor S₀ ofa sample 0 and the void content p₀ or the sample 0. The void factor ofthe sample 2 or a sample having a larger number is expressed by thescale factor of the sample in question, the scale fact or S₀ of a sample0 and the void content p₀ of the sample 1. After all, the void contentof each sample is expressed by the void content p₀ of only one sample,e.g., sample 0, through the mediation of the scale factor S. This meansthat if the scale factors S of the samples have been determined and thevoid content p₀ of one sample is determined, the void contents of othersamples can be easily calculated.

Referring to formula (17), the matrix density ρ_(M1) of a sample 1 isexpressed by the scale factor S₁ of the sample 1, the average densityρ_(F1) of the sample 1, the scale factor S₀ of the sample 0, the averagedensity ρ_(F0) of the sample 0, and the matrix density ρ_(M0) of asample 0. The matrix density of the sample 2 or a sample having a largernumber is similarly expressed. Namely, the matrix density of each sampleis expressed by the matrix density ρ_(M0) of only any one sample, e.g.,sample 0, through the mediation of the scale factor S and the averagedensity ρ_(F). This means that if the scale factors S and the averagedensities ρ_(F) of the samples are determined and the matrix densityρ_(M0) of any one sample is determined, the matrix densities of othersamples can be easily calculated.

In formula (17), unknown items are matrix densities ρ_(M0), ρ_(M1),ρ_(M2), ρ_(M3), . . . and the number of the unknown items is equal tothe number of the samples, while the number of the formulas in questionis less than the number of the samples by one. Therefore, the provisionsare insufficient to solve formula (17). Then, it is assumed that thematrix densities of the plural samples are identical with each other. Inother words, such samples should be the objects to be measured. In thiscase, formula (17) can be solved by adding the provision of “thedifference in matrix density among the samples becomes a minimum”. Sucha provision can be expressed by formula (18) in FIG. 5. The left side ofthe formula (18) represents what should be a minimum in connection withthe matrix densities ρ_(M0), ρ_(M1), ρ_(M2), . . . and uses the functionname of “MinFun”. The right side shows the form of the function, thenumerator being a square root of the sum, for all of the combinations oftwo samples, of the square of the difference between the matrix densityρ_(Mi) of the ith sample and the matrix density ρ_(Mj) of the jthsample, while the denominator being the sum of the matrix densities ofall of the samples. It is noted that each of the matrix densities of thesamples becomes a function of the matrix density ρ_(M0) of one sample asshown in formula (17) and thus formula (18) becomes a function of thematrix density ρ_(M0) only. Then, when differentiating formula (18) withthe matrix density ρ_(M0) and allowing the resultant to be zero as shownin formula (19), this operation would satisfy the provision that thedifference in matrix density among the samples becomes a minimum. Whenthe formula (19) is solved, the matrix density ρ_(M0) of the sample 0 isdetermined. When the matrix density ρ_(M0) is determined, the matrixdensities ρ_(M1), ρ_(M2), ρ_(M3), . . . of the other samples can becalculated based on formula (17). When the matrix density ρ_(M) of eachsample is determined, the void content p can be calculated based onformula (2). The description above is the principle of calculation ofthe void content according to the second aspect of the presentinvention.

Furthermore, when the void content is calculated, the equipment constantk₀ can be determined using formula (6) in FIG. 2. When the equipmentconstant k₀ is determined, thereafter, as in the first aspect, the voidcontent can be calculated based on formula (7) in FIG. 2 even for thesample having a different matrix density as long as the scale factor Sis determined.

Explaining an example of calculation of formula (19), the case usingthree samples will be described below. The matrix density Σ_(M0) of thesample 0 becomes formula (20) in FIG. 6. Namely, the matrix densityρ_(M0) of the sample 0 can be calculated with the use of the scalefactors S₀, S₁ and S₂ and the average densities ρ_(F0), ρ_(F1) andρ_(F2) of the samples 0, 1 and 2. Each of the average densities of thesamples can be measured by the X-ray reflectance method, and each of thescale factors can be determined by the X-ray small angle scatteringmethod. When the matrix density ρ_(M0) is determined, the matrixdensities ρ_(M1) and ρ_(M2) can be calculated using formulae (21) and(22), thus the matrix densities of the three samples can be determined.

A method for determining a particle content according to the secondaspect will now be described. A theoretical profile of the scatteredintensity is produced, for plural samples, with the use of formula (9)in FIG. 3 including the particle density ρ_(par), and a fittingoperation is carried out between the measured profile of the scatteredintensity and the theoretical profile to determine the scale factor S.On the other hand, the matrix density of each sample can be expressed byformula (23) in FIG. 6, which corresponds to formula (17) for the voids.The matrix density ρ_(M1) of the sample 1 is expressed by the scalefactor S₁ of the sample 1, the average density ρ_(F) of the sample 1,the particle density ρ_(par1) of the sample 1, the scale factor S₀ ofthe sample 0, the average density ρ_(F) of the sample 0, the particledensity ρ_(par0) of the sample 0 and the matrix density ρ_(M0) of thesample 0. The matrix density of the sample 2 or a sample having a largernumber is expressed similarly. After all, the matrix density of eachsample is expressed by the matrix density ρ_(M0) of only one sample,e.g., sample 0, through the mediation of the scale factor S, the averagedensity ρ_(F) and the particle density ρ_(par). This means that if thescale factors S, the average density ρ_(F) and the particle densityρ_(par) of the samples have been determined and the matrix densityρ_(M0) of one sample is determined, the matrix densities of othersamples can be easily calculated.

As in the case having the voids, the matrix density ρ_(M0) can becalculated by solving formula (19) in FIG. 5. When the matrix densityρ_(M0) is determined, the matrix densities ρ_(M1), ρ_(M2), ρ_(M3), . . .of the other samples can be calculated based on formula (23). When thematrix density ρ_(M) of each sample is determined, the particle contentp can be calculated based on formula (24) in FIG. 6, which is thetransformation of formula (34) in FIG. 20. The description above is theprinciple of calculation of the particle content according to the secondaspect of the present invention.

In the second aspect described above, when there is no thin film havinga known void content or particle content, the equipment constant isdetermined in a manner that there are prepared a plurality of sampleshaving void or particle contents which are different from each otherunder the provision that “the particle density is known”, and the matrixdensities are determined so that the matrix densities are identical witheach other to determine the equipment constants. On the contrary, it maybe contemplated that the particle density is unknown while “the matrixdensity is known”. In such a case, the third aspect should be used asdescribed below.

In the third aspect, a theoretical profile of the scattered intensity isproduced, for plural samples, with the use of formula (28) in FIG. 20including the matrix density ρ_(M), and a fitting operation is carriedout between the measured profile of the scattered intensity and thetheoretical profile to determine the scale factor S. In this case,formulae (33) and (35) are effected, and the particle density of eachsample is expressed by formula (36) in FIG. 21, which corresponds toformula (23) for the second aspect. The particle density ρ_(par0) can bedetermined so that the difference in particle density becomes a minimumby solving the minimum provision, for the particle density ρ_(par),similar to formula (19) in FIG. 5. When the particle density ρ_(par0) isdetermined, the particle densities ρ_(par1), ρ_(par1), . . . of theother samples can be calculated based on formula (36) Further, when theparticle density ρ_(par) of each sample is determined, the particlecontent can be calculated based on formula (24). The description aboveis the principle of calculation of the particle content according to thethird aspect of the present invention.

The scattering function F in formula (3) will now be described. Thescattering function may be one of some functions, typically thefunctions shown by formulae (25) and (26) in FIG. 18. Formula (25) showsa model which uses the average diameter D₀ of the void or particle andthe variance σ indicating the distribution of the diameter, whereQ(D,D₀,σ) is a diameter distribution function of the void or particle,the D being a variable representing the diameter. On the other hand,formula (26) shows a Debye model which uses the correlation distance ξof the electron density fluctuation. The selection of the models dependson the state of the voids or particles in the sample. Formula (25) is amodel function effective in a system in which the void or particle has aspecific shape, a sphere for example. On the contrary, formula (26) isknown as a model function, a type of two-layer separation, effective ina system in which the voids or particles are confusing like an ant nest.The embodiment described below uses the model function of formula (25)to determine the theoretical profile of the scattered intensity.

The average atomic scattering factor f_(F) and the average atomic massM_(F) appearing in formula (3) will now be described. The atomicscattering factor f can be calculated by formula (27) in FIG. 19 and isexpressed by a factor f₀ which does not depend on the wavelength andabnormal dispersions f₁ and f₂ which depend on the wavelength. Thesymbol i in the formula represents an imaginary number. The factor f₀depends on the scattering angle and is substantially equal to the atomicnumber Z in the small angle scattering region. The embodiment describedbelow uses, as a sample, MSQ (methyl silsesquioxane, chemical formulabeing Si₂O₃C₂H₆). The average atomic scattering factor f_(F) and theaverage atomic mass M_(F) of the MSQ are shown in a Table in FIG. 19,the X-ray wavelength being assumed to be CuKα, 0.154178 nm.

Actual measurement examples will now be described. Examples of voidcontent measurement according to the second aspect of the presentinvention will be shown for three porous MSQ thin films which areexpected to have different void contents. The three samples are expectedto have the same but unknown matrix density. First, an X-ray reflectanceprofile was measured for each of the three samples, called as samples 0,1 and 2, the method of the measurement being described briefly. FIG. 7Ashows the X-ray reflectance measurement. An X-ray 22 is incident on thesurface of a sample 20 at a minute incident angle θ. A reflected X-ray24 is detected in a direction of an outgoing angle θ, the same as theincident angle, from the surface of the sample 20. Assuming that theposition of the incident X-ray 22 is stationary, the sample is rotatedwith a θ-rotation and the X-ray detector, i.e., the direction of thereflected X-ray 24, is rotated with a 2θ-rotation, so that a variationof an X-ray reflectance, i.e., a ratio of a reflected X-ray intensity toan incident X-ray intensity, is recorded to obtain an X-ray reflectanceprofile.

FIG. 8 is a graph showing the measurement result of the X-rayreflectance profiles for the three samples, 2θ in abscissa and areflected X-ray intensity in ordinate. A solid line represents a profileof the sample 0, a broken line represents a profile of the sample 1 anda chain line represents a profile of the sample 2, noting however thatthe profiles for the samples 1 and 2 shown are limited, for avoidingcomplication, to only the starting regions in which the reflectancebegins to decrease. It would be readily expected with the graph that thevoid contents of the samples are different from each other because thecritical angles of the total reflection, i.e., the angle at which thereflectance begins to decrease, of the three samples are different fromeach other. A parameter fitting operation is carried out between themeasured X-ray reflectance profile and the theoretical X-ray reflectanceprofile to determine the film thickness and the density of the sampleand the roughness of the boundary between the thin film and thesubstrate, the parameter fitting operation being known, as disclosed inthe third publication mentioned above, and detailed explanation thereofbeing omitted. FIG. 9 shows a table indicating parameters determined bythe fitting operation. Now, the average density and the roughness havebeen determined for the three samples. It is seen at least that the voidcontents of the three samples would be different from each other becauseof the same matrix material and the different average densities.

Next, an X-ray small angle scattering measurement was carried out foreach of the three samples. An incident X-ray intensity profile wasmeasured for each sample for confirmation on the occasion of the X-raysmall angle scattering measurement for the three samples, the resultbeing shown in FIG. 10. The graph in FIG. 10 shows three incident X-rayintensity profiles superimposed on each other, but shows one curvebecause these profiles are perfectly superimposed. It is understood withthe graph that incident X-ray intensities are identical with each otherin the X-ray small angle scattering measurement for the three samples.The X-ray small angle scattering profiles were measured under theseconditions.

There will now be explained an offset scanning profile and a rockingscanning profile in the X-ray small angle scattering method. FIG. 7Bshows a method for measuring the offset scanning profile. An X-ray 22 isincident on the surface of a sample 20 at a minute incident angle θ. Ascattered X-ray 26 is detected in a direction of an outgoing angle“θ+Δθ” from the surface of the sample 20. Namely, the outgoing angle isoffset by Δθ compared to the incident angle. The offset allows theintense total-reflected X-ray not to enter the X-ray detector. Assumingthat the position of the incident X-ray 22 is stationary, the sample isrotated with a θ-rotation and the X-ray detector, i.e., the direction ofthe scattered X-ray 26, is rotated with a 2θ-rotation, so that avariation of the scattered X-ray intensity is recorded to obtain anoffset scanning profile. In general, the profile is measured in a rangebetween 0 to 8 degrees in 2θ.

FIG. 7C shows a method for measuring the rocking scanning profile. AnX-ray 22 is incident on the surface of a sample 20 at a minute incidentangle ω. A scattered X-ray 26 is detected in a direction which is a 2θagainst the incident X-ray 22 and is set stationary. Assuming that theposition of the incident X-ray 22 is stationary, only the sample isrotated with an ω-rotation, so that a variation of the scattered X-rayintensity is recorded to obtain a rocking scanning profile.

FIG. 11 is a graph showing the offset scanning profiles of the X-raysmall angle scattering for the three samples. The offset angle Δθ is 0.1degree. A profile fitting, i.e., the parameter fitting, operation iscarried out, for the three measured profiles, between the measuredprofile and the theoretical profile.

FIG. 12 is a graph showing the profile fitting on the offset scanningprofile of the X-ray small angle scanning for the sample 1. The offsetangle Δθ is 0.1 degree. Small circles indicate the measured value andthe assembly thereof becomes the measure profile. The chain linerepresents the theoretical profile of the scattered X-ray intensitycaused by the voids, the intensity being determined using formulae (37)to (41). The broken line represents the theoretical profile of thescattered X-ray intensity caused by the roughness of the surfaceboundary, the intensity being calculated using known theoretical formulaas disclosed in the second publication mentioned above. The solid linerepresents the sum of the two theoretical profiles. The parameters arechanged so that the total theoretical profile approaches the measuredprofile as close as possible to select the optimum parameters. If thescattered intensity caused by the voids is far larger than the scatteredintensity caused by the roughness, the scattered intensity caused by theroughness may be omitted in the fitting operation. The embodiment uses,as the scattering function, a model function of formula (25) mentionedabove, in which spherical voids are randomly dispersed in the thin filmand the distribution of the void size conforms to the Gamma distributionfunction.

FIG. 13 is a graph showing the profile fitting on the rocking scanningprofile of the X-ray small angle scanning for the sample 1, the angle 2θbeing 0.8 degree. It should be noted that the large measured peak at thecenter of the graph is caused the total reflection and thus not thescattered X-ray caused by the voids. The total reflection peak should beignored in the profile fitting operation with the theoretical profile.

In the actual measurements, the offset scanning profile fittingoperation as shown in FIG. 12 was carried out for each of the threesamples and then the rocking scanning profile fitting operation as shownin FIG. 13 was carried out for each of the three samples. The rockingscan profile fitting operations were carried out with 0.6, 0.8, 1.0, 1.2and 1.5 degrees in 2θ. The profile fitting operations should be carriedout ideally at the same time between the measured values and thetheoretical values for all of the offset scanning profiles and therocking scanning profiles in a manner that the parameters are determinedso that the difference becomes a minimum. The method of nonlinear leastsquares is effectively used for the minimization of the difference.

As a result of the fitting mentioned above, the average diameter of thevoid and the variance indicating the diameter distribution, the varianceof the Gamma distribution function, were determined as the parameters asshown in FIG. 14 and further the scale factor S was determined from theX-ray Intensity on that occasion.

A method for calculating the matrix density from the scale factor S andthe average density ρ_(F) will now be described. The average densityρ_(F) has been determined and is shown in FIG. 9. The scale factor S hasbeen determined in the X-ray small angle scattering method and is shownin FIG. 14. Then, these values are substituted into formula (20) in FIG.5 to calculate the matrix density ρ_(M0) of the sample 0, the resultantvalue being 1.412. Further, the matrix densities ρ_(M1) and ρ_(M2) ofthe sample 1 and sample 2 are calculated using formulae (21) and (22) inFIG. 6, the resultant values being 1.413 and 1.412. The void content pis calculated using the matrix densities in formula (2) in FIG. 2, thevoid contents of the three samples being p₀=0.175, p₁=0.237 and p₂=0.296respectively. These results are shown in the table in FIG. 17.

The minimization of the difference in matrix densities among the sampleswill now be described from another viewpoint. FIG. 15 is a graph showingrelationships of formula (17) in connection with the matrix densities ofthe three samples, the matrix density ρ_(M0) of the sample 0 in abscissaand the matrix densities ρ_(M0), ρ_(M1) and ρ_(M2) of the three samplesin ordinate. The average densities ρ_(F0), ρ_(F1) and ρ_(F2) and thescale factors S₀, S₁ and S₂ in formula (17) receive the values shown inthe table in FIG. 17. Thinking about the sample 0 and the sample 1, thematrix densities of the samples become identical with each other at theintersection of the two straight lines. The matrix density at theintersection is the value which allows the difference in the two matrixdensities to be a minimum, i.e., zero. Accordingly, if at least twosamples are measured, the matrix density can be determined.

Furthermore, thinking about the three samples, three intersections existamong the three straight lines. It is noted however that the graph shownin FIG. 15 the three straight lines are intersected with each other atalmost the same point, the matrix density at the intersection being1.412. The fact that three intersections are at almost the same pointrepresents a high reliability of the matrix density determined by thepresent invention. If the three intersections are apart from each other,the reliability of the matrix density calculated using formula (20)would be poor. Thus, it would be preferable to carry out the secondaspect for at least three samples as shown in FIG. 15 for confirming thereliability of the matrix density.

FIG. 16 is a graph showing the relationships of formula (16) inconnection with the void densities of the three samples. It isunderstood that when the void content p₀ of the sample 0 is determined,the void contents of the sample 1 and the sample 2 are easilycalculated.

Although the average density and the film thickness of the sample aredetermined by the X-ray reflectance method in the embodiments describedabove, these values may be acquired by another means, for example, theymay be entered via a keyboard by an operator.

1. A data processing apparatus for measuring a void content of a samplewhich is made of a thin film having a matrix and voids dispersed in thematrix, the apparatus comprising: (a) film thickness acquiring means foracquiring a film thickness of the sample; (b) average density acquiringmeans for acquiring an average density of the sample; (c) an X-ray smallangle scattering measurement equipment for acquiring a measured profileof a scattered X-ray intensity of the sample; (d) theoretical profileproducing means for producing a theoretical profile of the scatteredX-ray intensity of the sample using the acquired film thickness and theacquired average density; (e) scale factor calculating means forcarrying out a parameter fitting operation between the acquired measuredprofile of the scattered X-ray intensity and the produced theoreticalprofile of the scattered X-ray intensity, and for calculating a scalefactor of the scattered X-ray intensity; (f) void content acquiringmeans for acquiring a known void content of a first sample; (g)equipment constant calculating means for calculating an equipmentconstant of the X-ray small angle scattering measurement equipment basedon the known void content of the first sample and a first scale factorcalculated by the scale factor calculating means for the first sample;(h) void content calculating means for calculating a void content of asecond sample for which the void content is unknown based on a secondscale factor calculated by the scale factor calculating means for thesecond sample and the calculated equipment constant; and (i) signalgenerating means for generating a signal indicative of the calculatedvoid content of the second sample.
 2. A data processing apparatus formeasuring a particle content of a sample which is made of a thin filmhaving a matrix and particles, with a known particle density, dispersedin the matrix, the apparatus comprising: (a) film thickness acquiringmeans for acquiring a film thickness of the sample; (b) average densityacquiring means for acquiring an average density of the sample; (c)particle density acquiring means for acquiring the particle density; (d)an X-ray small angle scattering measurement equipment for acquiring ameasured profile of a scattered X-ray intensity of the sample; (e)theoretical profile producing means for producing a theoretical profileof the scattered X-ray intensity of the sample using the acquired filmthickness, the acquired average density, and the acquired particledensity; (f) scale factor calculating means for carrying out a parameterfitting operation between the acquired measured profile of the scatteredX-ray intensity and the produced theoretical profile of the scatteredX-ray intensity, and for calculating a scale factor of the scatteredX-ray intensity; (g) particle content acquiring means for acquiring aknown particle content of a first sample; (h) equipment constantcalculating means for calculating an equipment constant of the X-raysmall angle scattering measurement equipment based on the known particlecontent of the first sample and a first scale factor calculated by thescale factor calculating means for the first sample; (i) particlecontent calculating means for calculating a particle content of a secondsample for which the particle content is unknown based on a second scalefactor calculated by the scale factor calculating means for the secondsample and the calculated equipment constant; and (j) signal generatingmeans for generating a signal indicative of the calculated particlecontent of the second sample.
 3. A data processing apparatus formeasuring a particle content of a sample which is made of a thin filmhaving a matrix with a known matrix density and particles dispersed inthe matrix, the apparatus comprising: (a) film thickness acquiring meansfor acquiring a film thickness of the sample; (b) average densityacquiring means for acquiring an average density of the sample; (c)matrix density acquiring means for acquiring the matrix density; (d) anX-ray small angle scattering measurement equipment for acquiring ameasured profile of a scattered X-ray intensity of the sample; (e)theoretical profile producing means for producing a theoretical profileof the scattered X-ray intensity of the sample using the acquired filmthickness, the acquired average density and the acquired matrix density;(f) scale factor calculating means for carrying out a parameter fittingoperation between the acquired measured profile of the scattered X-rayintensity and the produced theoretical profile of the scattered X-rayintensity, and for calculating a scale factor of the scattered X-rayintensity; (g) particle content acquiring means for acquiring a knownparticle content of a first sample; (h) equipment constant calculatingmeans for calculating an equipment constant of the X-ray small anglescattering measurement equipment based on the known particle content ofthe first sample and a first scale factor calculated by the scale factorcalculating means for the first sample; (i) particle content calculatingmeans for calculating a particle content of a second sample for whichthe particle content is unknown based on a second scale factorcalculated by the scale factor calculating means for the second sampleand the calculated equipment constant; and (j) signal generating meansfor generating a signal indicative of the calculated particle content ofthe second sample.
 4. A data processing apparatus for measuring a voidcontent of at least one of plural samples which is made of a thin filmhaving a matrix and voids dispersed in the matrix, the apparatuscomprising: (a) film thickness acquiring means for acquiring a filmthickness of the sample; (b) average density acquiring means foracquiring an average density of the sample; (c) an X-ray small anglescattering measurement equipment for acquiring a measured profile of ascattered X-ray intensity of the sample; (d) theoretical profileproducing means for producing a theoretical profile of the scatteredX-ray intensity of the sample using the acquired film thickness and theacquired average density; (e) scale factor calculating means forcarrying out a parameter fitting operation between the acquired measuredprofile of the scattered X-ray intensity and the produced theoreticalprofile of the scattered X-ray intensity, and for calculating a scalefactor of the scattered X-ray intensity; (f) matrix density calculatingmeans for calculating matrix densities of the plural samples based onscale factors calculated for the plural samples by the scale factorcalculating means so that differences in the matrix densities among theplural samples become a minimum; (g) void content calculating means forcalculating the void content of the at least one of the plural samplesbased on the acquired average density and the calculated matrix densityof the at least one of the plural samples; and (h) signal generatingmeans for generating a signal indicative of the calculated void contentof the at least one of the plural samples.
 5. A data processingapparatus for measuring a particle content of at least one of pluralsamples which is made of a thin film having a matrix and particles, witha known particle density, dispersed in the matrix, the apparatuscomprising: (a) film thickness acquiring means for acquiring a filmthickness of the sample; (b) average density acquiring means foracquiring an average density of the sample; (c) particle densityacquiring means for acquiring the particle density; (d) an X-ray smallangle scattering measurement equipment for acquiring a measured profileof a scattered X-ray intensity of the sample; (e) theoretical profileproducing means for producing a theoretical profile of the scatteredX-ray intensity of the sample using the acquired film thickness, theacquired average density and the acquired particle density; (f) scalefactor calculating means for carrying out a parameter fitting operationbetween the acquired measured profile of the scattered X-ray intensityand the produced theoretical profile of the scattered X-ray intensity,and for calculating a scale factor of the scattered X-ray intensity; (g)matrix density calculating means for calculating matrix densities of theplural samples based on scale factors calculated for the plural samplesby the scale factor calculating means so that differences in the matrixdensities among the plural samples become a minimum; (h) particlecontent calculating means for calculating the particle content of the atleast one of the plural samples based on the acquired average density,the acquired particle density and the calculated matrix density of theat least one of the plural samples; and (i) signal generating means forgenerating a signal indicative of the calculated particle content of theat least one of the plural samples.
 6. A data processing apparatus formeasuring a particle content of at least one of plural samples which ismade of a thin film having a matrix with a known matrix density andparticles dispersed in the matrix, the apparatus comprising: (a) filmthickness acquiring means for acquiring a film thickness of the sample;(b) average density acquiring means for acquiring an average density ofthe sample; (c) matrix density acquiring means for acquiring the matrixdensity; (d) an X-ray small angle scattering measurement equipment foracquiring a measured profile of a scattered X-ray intensity of thesample; (e) theoretical profile producing means for producing atheoretical profile of the scattered X-ray intensity of the sample usingthe acquired film thickness, the acquired average density and theacquired matrix density; (f) scale factor calculating means for carryingout a parameter fitting operation between the acquired measured profileof the scattered X-ray intensity and the produced theoretical profile ofthe scattered X-ray intensity, and for calculating a scale factor of thescattered X-ray intensity; (g) particle density calculating means forcalculating particle densities of the plural samples based on scalefactors calculated for the plural samples by the scale factorcalculating means so that differences in the particle densities amongthe plural samples become a minimum; (h) particle content calculatingmeans for calculating the particle content of the at least one of theplural samples based on the acquired average density, the acquiredmatrix density and the calculated particle density of the at least oneof the plural samples; and (i) signal generating means for generating asignal indicative of the calculated particle content of the at least oneof the plural samples.